3.7 Cylindrical and Spherical Coordinates

]> 3.7 Cylindrical and Spherical Coordinates

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3.7 Cylindrical and Spherical Coordinates

In three dimensions there are two analogues of polar coordinates.

In cylindric coordinates, $x$ and $y$ are described by $r$ and $θ$ exactly as in two dimensions, while the third dimension, $z$ is treated as an ordinary coordinate.
$r$ then represents distance from the $z$ axis.

In spherical coordinates, a general point is described by two angles and one radial variable, $ρ$ , which represents distance to the origin: $ρ^{2} = x^{2} + y^{2} + z^{2}$ .

The two angular variables are related to longitude and latitude, but latitude is zero at the equator, and the variable $ϕ$ that we use is 0 on the $z$ axis (which means at the north pole).

We define $ϕ$ by $\cos ϕ = \frac{z}{ρ}$ , so that with $r$ defined as always here by $r^{2} = x^{2} + y^{2}$ , we have $\sin ϕ = \frac{r}{ρ}$ .

The longitude angle $θ$ is defined by $\tan θ = \frac{y}{x}$ , exactly as in two dimensions. We therefore have $x = r \cos θ = ρ \sin ϕ \cos θ$ , and what is $y$ ?

Exercises:

3.12 Express the parameters of cylindric and spherical coordinates in terms of $x, y$ and $z$ .

3.13 Construct a spreadsheet converter which takes coordinates $x, y$ and $z$ and produces the three parameters of spherical coordinates; and vice versa. Verify that they work by substituting the result from one as input into the other.